A monkey is swinging from a tree. On the first swing, she passes through an arc of $20\text{ m}$. With each swing, she passes through an arc $\dfrac{4}{5}$ the length of the previous swing. What is the total distance the monkey has traveled when she completes her $10^\text{th}$ swing? Round your final answer to the nearest meter.
Answer: Notice that the lengths of the monkey's swings form a geometric sequence. The total distance traveled after $ n$ swings is the ${\text{sum}}$ of the first $n$ terms in the sequence. This is called a geometric series. This is the formula for that sum: $ S={a}\left(\dfrac{1-{r}^{ n}}{1-{r}}\right)$ where ${a}$ is the first term and ${r}$ is the common ratio. We can use this formula, along with the given information, to find the value of the sum, $ S$. Using the given information We are given that the length of the first swing is ${20}\text{ m}$. This is the first term $ a$. We are given that each swing is ${\dfrac{4}{5}}$ the length of the previous swing. This is the common ratio $ r$. There are ${10}$ swings in the series. This is the number of terms $ n$. We want to find the total distance. This is the sum $ S$. Finding the sum $\begin{aligned} S&={20} \cdot \dfrac{1-\left({\dfrac{4}{5}}\right)^{{10}}}{1-\left({\dfrac{4}{5}}\right)} \\\\ &\approx{20} \cdot \dfrac{0.8926}{0.2} \\\\ & \approx89.26\,\text{m} \end{aligned}$ Answer To the nearest meter, the total distance the monkey has traveled when she completes her $10^\text{th}$ swing is $89\,\text{m}$.